
Using (REF ) with \(\alpha _{eff}=1/(16 \pi G_{eff})\)  we obtain that \(\delta = 8 \pi \, G_{eff}\,M_{flat}\)  which is the formula for the angular deficit produced by a mass \(M_{flat}\)  in \(2+1\)  Minkowski spacetime [1]} if \(G_{eff}\)  replaces \(G\)  in [1]}. Asymptotically, the spacetime is locally flat but topologically a cone. There is however no conical singularity at the origin in our case in contrast to the point mass in [1]}. The spacetime is smooth at the origin since the vortex by construction is an extended non-singular object. In our case, the conical spacetime is only the asymptotic spacetime and does not extend into the core of the vortex.
